Black Scholes
Black-Scholes Option Pricing Model
Nathan Coelen
June 6, 2002
1
Introduction
Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change, modern financial instruments have become extremely complex. New mathematical models are essential to implement and price these new financial instruments. The world of corporate finance once managed by business students is now controlled by mathematicians and computer scientists.
In the early 1970’s, Myron Scholes, Robert Merton, and Fisher Black made an important breakthrough in the pricing of complex financial instruments by developing what has become known as the Black-Scholes model. In 1997, the importance of their model was recognized world wide when Myron Scholes and Robert Merton received the Nobel Prize for Economics. Unfortunately,
Fisher Black died in 1995, or he would have also received the award [Hull,
2000]. The Black-Scholes model displayed the importance that mathematics plays in the field of finance. It also led to the growth and success of the new field of mathematical finance or financial engineering.
In this paper, we will derive the Black-Scholes partial differential equation and ultimately solve the equation for a European call option. First, we will discuss basic financial terms, such as stock and option, and review the arbitrage pricing theory. We will then derive a model for the movement of a stock, which will include a random component, Brownian motion. Then, we will discuss some basic concepts of stochastic calculus that will be applied to our stock model. From this model, we will derive the Black-Scholes partial differential equation, and I will use boundary conditions for a European call option to solve the equation.
1
2
Definitions
Financial assets are claims on some issuer, such as the federal government or a corporation, such as Microsoft. Financial assets also include real assets such as real
Nathan Coelen
June 6, 2002
1
Introduction
Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change, modern financial instruments have become extremely complex. New mathematical models are essential to implement and price these new financial instruments. The world of corporate finance once managed by business students is now controlled by mathematicians and computer scientists.
In the early 1970’s, Myron Scholes, Robert Merton, and Fisher Black made an important breakthrough in the pricing of complex financial instruments by developing what has become known as the Black-Scholes model. In 1997, the importance of their model was recognized world wide when Myron Scholes and Robert Merton received the Nobel Prize for Economics. Unfortunately,
Fisher Black died in 1995, or he would have also received the award [Hull,
2000]. The Black-Scholes model displayed the importance that mathematics plays in the field of finance. It also led to the growth and success of the new field of mathematical finance or financial engineering.
In this paper, we will derive the Black-Scholes partial differential equation and ultimately solve the equation for a European call option. First, we will discuss basic financial terms, such as stock and option, and review the arbitrage pricing theory. We will then derive a model for the movement of a stock, which will include a random component, Brownian motion. Then, we will discuss some basic concepts of stochastic calculus that will be applied to our stock model. From this model, we will derive the Black-Scholes partial differential equation, and I will use boundary conditions for a European call option to solve the equation.
1
2
Definitions
Financial assets are claims on some issuer, such as the federal government or a corporation, such as Microsoft. Financial assets also include real assets such as real
Bibliography: [Baxter, 1996] Baxter, Martin and Andrew Rennie. Financial Calculus: An Introduction to Derivative Pricing University Press, 1996. [Hull, 2000] Hull, John C. Options, Futures, and Other Derivatives. Upper Saddle River, New Jersey: Prentice Hall, 2000. [Klebaner, 1998] Klebaner, Fima C. Introduction to Stochastic Calculus with Applications [Ross, 1999] Ross, Sheldon. An Introduction to Mathematical Finance. Cambridge, England: Cambridge University Press, 1999. [Wilmott, 1995] Wilmott, Paul, Sam Howison, and Jeff Dewynne. The Mathematics of Financial Derivatives Press, 1995.